If you connect RB directly to ISP instead of Cisco, then of course it won't work without some config changes. Did you perhaps miss what I was trying to tell you?
Once again, with the original setup you have two options (if you want to have separate subnet):
a) On Cisco, add static route to 192.168.0.0/24 with gateway 192.168.1.114. If you do this, internet access from 192.168.0.0/24 will work. Also most likely computers from 192.168.0.0/24 and 192.168.1.0/24 will be able to communicate with each other (but it also depends on Cisco).
b) On RB, enable NAT on WAN port. This way, all connections from 192.168.0.0/24 will be hidden behind common address 192.168.1.114 and no configuration will be necessary elsewhere. But you'll have double NAT, which is not the best thing.
People who quote full posts should be spanked with ethernet cable. Some exceptions for multi-topic threads may apply. Not intended as incentive for masochists.